A brake system for motor vehicles is known from DE 10 2010 040 097 A1, which is incorporated by reference. The previously known brake system is designed as an “open system”, in which an excess of pressure medium arising during wheel brake pressure modulation processes is passed to the pressure medium reservoir via a hydraulic return line. The disadvantage with open brake systems is that, if there is a leak in an outlet valve, the brake circuit concerned fails. Another disadvantage of the known arrangement is that, owing to this risk of failure, it is only possible to use seat valves acted upon in the closing direction by the wheel brake pressures as outlet valves. As a result, only digitally controlled seat valves can be used as outlet valves, and it is not possible to achieve infinitely variable, comfortable and quiet pressure control with these valves. Analog-controlled, wheel-specific pressure reduction is not possible with the outlet valves of the known arrangement because operation of a seat valve under analog control presupposes the opposite direction of flow. The same argument also excludes use of the inlet valves for wheel-specific pressure reduction control in the direction of the pressure medium supply device. Because these inlet valves capable of analog control in the pressure buildup direction cannot be controlled in the direction of flow for pressure reduction and would be “pulled shut”, a check valve that opens in the direction of pressure reduction is connected in parallel with each inlet valve. Thus, in the known arrangement, the pressure medium supply device must always supply a pressure which is at least as high as the highest of the wheel brake pressures demanded. A pressure buildup carried out with the controllable pressure medium supply device is possible only for the wheel brake with the highest instantaneously demanded pressure, while the remaining, lower wheel pressures are derived from the maximum pressure via the inlet valves capable of analog control. However, this leads to wheel brake characteristics with a lower quality of control if the differences between the required wheel pressures and the maximum pressure are small. With predetermined setpoint wheel brake pressures, the pressure to be output by the pressure medium supply device is fixed with this pressure control concept. At the same time, it is not possible to exploit the fact that it is simpler to provide a precise pressure medium volume than a precise pressure with a pressure and volume setting device. While it is a simple matter to output and accurately meter a required volume by means of the pressure and volume setting device, a required pressure is established by a method in which the actual pressure output is detected by means of a pressure sensor and a system pressure controller calculates and implements the change in pressure medium volume required to bring the actual pressure closer to the setpoint pressure. When the method described is used for wheel brake pressure control in the known arrangement, acceptable quality of control is difficult to achieve.